# Divisible abelian group

## Definition

A **divisible abelian group** is an abelian group satisfying the following equivalent conditions:

- For every and nonzero integer , there exists such that .
- Viewing the category of abelian groups as the category of modules over the rin of integers, is an injective module.

## Examples

- The group of rational numbers, and more generally, the additive group of any vector space over the field of rational numbers, is a divisible abelian group. In fact, it is a
*uniquely*divisible abelian group. - The group of rational numbers modulo integers is a divisible abelian group.
- The quasicyclic group for a prime , i.e., the group of all roots of unity for all under multiplication, is also a divisible abelian group.